Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of C3⋊(C3⋊AGL(1,𝔽7)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊AGL(1,𝔽7)) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3⋊D3) |
| Fitting, socle, derived | C32×C7 ⋊ C2×C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2 |
| Sylow | C7 ⋊ C3×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C7 | C2×C3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,3)(5,6)(8,11,12,9,10,13);(4,5,6)(7,8,10,13,12,11,9);(1,2,3)Polycyclic presentation:
2,3^3,7|c^a=c^2,d^a=d^2,e^a=e^6,e^b=e^227 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 51);